Re: [問題] 順序統計問題
※ 引述《Hshs (免錢沒人愛@@)》之銘言：
: Let X1 X2 X3 are the order statistics of the iid random variables X1 X2 and
: X3 with common exponential distribution with mean 1/λ. Show that
: Y1 = X3 - X2 and Y2 = X2 are independent
: 請問一下該怎麼解呢? 謝謝.
: 似乎是考古題,好像看過
f(x1,x2,x3)= 6 λ^3 exp (-λ(x1+x2+x3)) 0 < x1 < x2 < x3 < inf
f(x2,x3) = ∫6 λ^3 exp (-λ(x1+x2+x3)) dx1 (積分範圍0< x1 < x2)
f(x2,x3) = 6 λ^2 exp (-λx3) [exp (-λx2) - exp (- 2λx2) ] 0< x2 < x3 < inf
Let Y1= X3 - X2 and Y2 = X2 ==> X3 = Y1 + Y2 (0 < Y2 < Y1+Y2 < inf)
By Jacobian
f(Y1,Y2)= 6 λ^2 exp (-λ(Y1+Y2)) [exp (-λY2) - exp (- 2λY2) ] * 1
= 6 λ^2 exp (-λY1) [exp (-2λY2) - exp (- 3λY2) ]
積分範圍可以拆成 0< Y1 < inf 跟 0 < Y2 < inf
f(Y1,Y2) 可以拆成兩個 f(Y1) * f(Y2) 所以independent
--
不知道有沒有誤...
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 160.39.4.106
討論串 (同標題文章)
本文引述了以下文章的的內容：
完整討論串 (本文為第 2 之 2 篇)：